Vardaan
Class 9 Maths • Chapter 1 • NCERT + RS Aggarwal + RD Sharma

Number Systems

Vardaan Learning Institute  |  School-Exam Focused Notes

🔢 1. The Number System — Hierarchy

Real Numbers (ℝ) Rational Numbers (ℚ) — p/q, q≠0 Integers (ℤ) Whole Numbers (W) Natural Numbers (ℕ) 1, 2, 3, 4, 5 ... + 0 + Negatives + Fractions & Decimals e.g. ½, -¾, 0.25, 3.333... Irrational Numbers √2, √3, √5, π, e Non-terminating, non-repeating decimals π = 3.14159265... √2 = 1.41421356...
Type Definition Examples Symbol
Natural Numbers Counting numbers starting from 1 1, 2, 3, 4, 5... ℕ
Whole Numbers Natural numbers + zero 0, 1, 2, 3... W
Integers Whole numbers + negative numbers ...−2, −1, 0, 1, 2... ℤ
Rational Numbers Numbers of the form p/q where p,q∈ℤ and q≠0 ½, −¾, 0.25, 3, −7 ℚ
Irrational Numbers Non-terminating, non-repeating decimals. Cannot be written as p/q. √2, √3, π, e, √5 —
Real Numbers All rational + irrational numbers Every number on the number line ℝ

📌 2. Rational Numbers — Key Properties

Identifying Rational from its Decimal Expansion:
• Terminating decimal → Rational. Example: 0.25 = 25/100 = 1/4
• Non-terminating but repeating (recurring) → Rational. Example: 0.3̄ = 1/3, 0.142857142857... = 1/7
• Non-terminating, non-repeating → Irrational. Example: π = 3.14159..., √2 = 1.41421...

Converting Recurring Decimals to p/q Form (Very Important for Exams!)

Example 1: Convert 0.3̄ = 0.333... to p/q
Let x = 0.333...  →  10x = 3.333...
10x − x = 3.333... − 0.333...  →  9x = 3  →  x = 1/3

Example 2: Convert 0.47̄ = 0.4777... to p/q
Let x = 0.4777...  →  10x = 4.777...  →  100x = 47.777...
100x − 10x = 47.777... − 4.777... = 43  →  90x = 43  →  x = 43/90

Example 3: Convert 0.6̄7̄ = 0.676767... to p/q
Let x = 0.676767...  →  100x = 67.6767...
100x − x = 67  →  99x = 67  →  x = 67/99
📝 Exam Shortcut For 0.ā̄b̄... (k digits repeating from start): p/q = (repeating block) / (k nines)
e.g. 0.12̄3̄ → digits before decimal don't repeat → use subtraction method.

🔍 3. Irrational Numbers — Key Concepts

Important Results on Irrationals (RD Sharma / Board exams):
📐 Theorem (NCERT) — Prove √2 is irrational
Proof by Contradiction:
Assume √2 is rational. Then √2 = p/q where p, q are integers, q ≠ 0, and p/q is in lowest terms (HCF(p,q) = 1).
Squaring: 2 = p²/q²  →  p² = 2q²
So p² is even  →  p is even  →  p = 2m for some integer m
Substituting: (2m)² = 2q²  →  4m² = 2q²  →  q² = 2m² → q is even
But then both p and q are even, contradicting HCF(p,q) = 1.
∴ √2 is irrational.

Similarly √3, √5, √7 can be proved irrational. (Same method — important for board exams!)

📍 4. Representing Irrationals on Number Line

Method: Geometric construction (Spiral of Theodorus)

To represent √2 on a number line:
1. Draw OA = 1 unit on number line.
2. At A, draw AB ⊥ OA such that AB = 1 unit.
3. OB = √(OA² + AB²) = √(1+1) = √2 (by Pythagorean theorem).
4. With O as centre and OB as radius, draw arc to cut number line at C. OC = √2.

For √3: From √2 point, draw perpendicular of 1 unit. Hypotenuse = √3.
For √n: Continue the spiral — from √(n−1) point, draw perpendicular of 1 unit; hypotenuse = √n.
0 1 1 1 √2 √2 √3 √3 Blue △ → √2 Orange △ → √3

✖️ 5. Operations on Real Numbers

Laws of Radicals (Very Frequently Tested)

Law Rule Example
Product Rule √a × √b = √(ab) √3 × √5 = √15
Quotient Rule √a ÷ √b = √(a/b) √18 / √2 = √9 = 3
Power Rule (√a)² = a (√7)² = 7
aᵐ × aⁿ = aᵐ⁺ⁿ 3² × 3³ = 3⁵ = 243
aᵐ ÷ aⁿ = aᵐ⁻ⁿ 5⁶ ÷ 5² = 5⁴ = 625
(aᵐ)ⁿ = aᵐⁿ (2³)² = 2⁶ = 64
a⁰ = 1 (a ≠ 0) 7⁰ = 1
a⁻ⁿ = 1/aⁿ 2⁻³ = 1/8
a^(1/n) = ⁿ√a 8^(1/3) = ∛8 = 2
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ) 8^(2/3) = (∛8)² = 4

Rationalisation of Denominators (Very Important!)

The process of making the denominator free of radicals is called rationalisation.

Type 1 — Single radical in denominator:
1/√2 = 1/√2 × √2/√2 = √2/2

Type 2 — (a + √b) in denominator:
Multiply by conjugate: (a − √b)  →  uses identity (a+b)(a−b) = a²−b²
1/(3+√2) = (3−√2)/[(3+√2)(3−√2)] = (3−√2)/(9−2) = (3−√2)/7

Type 3 — (√a + √b) in denominator:
1/(√5+√3) = (√5−√3)/[(√5+√3)(√5−√3)] = (√5−√3)/(5−3) = (√5−√3)/2

Comparing Irrational Numbers

To compare √a and √b: compare a and b. √5 > √3 since 5 > 3.
To compare √a and n (a whole number): square both. Which is bigger, √7 or 2.5? → 7 vs 6.25 → √7 > 2.5
To compare n√a and m√b where n ≠ m: make indices same using LCM.
Example: 2^(1/2) vs 3^(1/3). LCM of 2,3 = 6. → 2^(3/6) vs 3^(2/6) → 8^(1/6) vs 9^(1/6) → 8 < 9 → 2^(1/2) < 3^(1/3)

📐 6. Finding Rational / Irrational Numbers Between Two Numbers

Finding rationals between a and b:
Method 1: Take average: (a+b)/2
Method 2: Multiply both by 10/100: find numbers of the form m/n
Example: Rationals between 1/3 and 1/2 = 5/12 (average), 2/5, 3/7, etc.

Finding irrationals between a and b:
Simply pick any √n where n is not a perfect square and a < √n < b
Example: Irrationals between 2 and 3: √5, √6, √7, √8 (since 4 < 5,6,7,8 < 9)
✏️ Practice Questions — Number Systems (35 Questions)

Section A — 1 Mark Questions Easy

Q1. Is 0 a natural number, whole number, or integer?
Q2. Identify: is 0.101001000... rational or irrational?
Q3. Express 0.6̄ as p/q.
Q4. Is √4 rational or irrational?
Q5. Simplify: (√3)²
Q6. True/False: π = 22/7. Justify.
Q7. Find a rational number between 3 and 4.
Q8. Evaluate: 125^(1/3)
Q9. Are all integers rational numbers?
Q10. Simplify: √50 (in simplest radical form)

Section B — 2/3 Mark Questions Medium

Q11. Convert 0.47̄ to p/q form.
Q12. Convert 1.2̄7̄ to p/q form.
Q13. Find 3 irrational numbers between 2 and 3.
Q14. Find 3 rational numbers between 2/3 and 3/4.
Q15. Rationalise: 1/(2+√3)
Q16. Rationalise: 5/(√7−√2)
Q17. Simplify: (2+√3)(2−√3)
Q18. Represent √5 on the number line.
Q19. Simplify: √72 + √50 − √32
Q20. Evaluate: (64)^(-1/3) × (64)^(1/3 + 2)

Section C — 3/4 Mark Questions (NCERT/Board Style) Medium

  1. Q21. Simplify: (5 + 2√3)/(7 + 4√3). Rationalise and simplify completely.
  2. Q22. If x = 2 + √3, find: (i) x + 1/x   (ii) x² + 1/x²
  3. Q23. Simplify: (√2 + √3)² + (√5 − √2)²
  4. Q24. If a = 3 + 2√2, find √a − 1/√a. Hence find a + 1/a.
  5. Q25. Simplify: [5^(1/2) × 5^(1/3)] ÷ 5^(1/6)
  6. Q26. Show that 5 − √3 is irrational (using proof by contradiction).
  7. Q27. Show that 3√2 is irrational.

Section D — 5 Mark / Long Answer Hard

  1. Q28. Prove that √2 is irrational. (Full proof — NCERT Appendix / Board question)
  2. Q29. If x = (√3+√2)/(√3−√2) and y = (√3−√2)/(√3+√2), find x² + y² + xy.
  3. Q30. Simplify: [(√3 + √2)/(√3 − √2)] − [(√3 − √2)/(√3 + √2)]
  4. Q31. Simplify: [3^(n+1) + 3^n] / [3^(n+2) − 3^(n+1)]
  5. Q32. Arrange in ascending order: 2^(1/2), 3^(1/3), 4^(1/4)
  6. Q33. Simplify: [(x^a / x^b)^(a+b)] × [(x^b / x^c)^(b+c)] × [(x^c / x^a)^(c+a)]
  7. Q34. If 2^x = 3^y = 12^z, prove that 1/z = 1/y + 2/x.
  8. Q35. Convert 2.3̄5̄ (2.353535...) to p/q form and verify by long division.
✅ Key Answers: Q3: 2/3 | Q8: 5 | Q10: 5√2 | Q11: 43/90 | Q12: 126/99=14/11 | Q15: 2−√3 | Q16: 5(√7+√2)/5 | Q17: 1 | Q19: 7√2 | Q22(i): 4 | Q25: 5^(7/6) | Q30: 4√6 | Q31: 1/2 | Q35: 233/99

📝 Quick Revision

  1. ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ. Every natural number is an integer, every integer is rational, every rational is real.
  2. Terminating or recurring decimal → Rational | Non-terminating, non-recurring → Irrational
  3. √p is irrational for any prime p. √4=2 is rational.
  4. Rationalise by multiplying by conjugate: 1/(a+√b) → multiply by (a−√b)/(a−√b)
  5. Key laws: aᵐ×aⁿ=aᵐ⁺ⁿ | aᵐ÷aⁿ=aᵐ⁻ⁿ | (aᵐ)ⁿ=aᵐⁿ | a^(1/n)=ⁿ√a | a^(m/n)=(ⁿ√a)ᵐ
  6. Infinite rationals and irrationals exist between any two real numbers